The multigraph linear-logarithmic expected-face conjecture

Let $G$ be an $n$-vertex multigraph with maximum edge-multiplicity $\mu$. Select an orientable embedding of $G$ uniformly at random, and let $F$ denote its number of faces. Multigraph expected-face conjecture. The expected number of faces satisfies

$$\mathbb{E}[F]=O\bigl(n\log(2\mu)\bigr).$$

This generalizes the simple-graph conjecture to multigraphs with parallel edges of bounded maximum multiplicity. The source states that this more general conjecture is treated, but the supplied text gives no resolution status.